Percentage Plateau

Question

We're trying to figure out something, and things aren't adding up. The senario is that you make $\$ 50,000$ a year. Every year you get a $15\%$ bonus of that income, which then gets added to your next year's income. So, the first year, you get $\$ 7,500$ which then makes your base $\$ 57,500$ the next year. When we try to figure out what the bonus will be past $6$ or $7$ years the numbers just sort of plateau at around $\$ 8,823$. All we were doing is adding the amount of the bonus to $\$ 50,000$ and then finding $15\%$ of the resulting number, and repeating. Is this correct? Why are the numbers plateauing? What would the bonus actually be after $6$ or $7$ years? I'm sorry if this is a simple question for this forum, it was recommended to me. Thanks so much in advance.

Answer 1

I am not sure this is what you want, but here is the calculation for the bonuses in the first $3$ years which should get you going:

1st Year:

$$ \text{Base} = 50,000 \quad \Rightarrow \quad \text{Bonus}=0.15*50,000 = 7,500. $$

2nd Year:

$$ \text{Base} = 50,000+7,500=57,500 \quad \Rightarrow \quad \text{Bonus}=0.15*57,500 = 8,625. $$

3rd Year:

$$ \text{Base} = 57,500+8,625=66,125 \quad \Rightarrow \quad \text{Bonus}=0.15*57,500 \approx 9,919. $$

Answer 2

The general rule is that in year $n$ the base is $50,000\cdot 1.15^n$ and the bonus is $7,500\cdot 1.15^n$ This increases without bound.

Answer 3

You start with a bonus of 7500. Then every year the bonus decreases about $85\%$. But you carry over the bonuses of the previous years. That means that at the first year we have $b_1=7500$. And in the second year $b_2=7500+0.15\cdot 7500=8625$ and so on:

$b_3=7500+0.15\cdot 7500+0.15^2\cdot 7500$

...

$b_n=7500\sum_{i=1}^{n} 0.15^{i-1}$

We can use the formula for the partial sum of a geometric series to calculate $b_n$

$$b_n=\frac{7500}{0.15}\sum_{i=1}^{n} 0.15^{i}=\frac{7500}{0.15}\cdot 0.15\cdot \frac{1-0.15^n}{1-0.15}=7500\cdot \frac{1-0.15^n}{1-0.15}$$

And for $$ n \to \infty, b_n=\frac{7500}{1-0.15}\approx 8823.53$$

This is the upper bound of b$_n$

The graph below shows that the increase of the bonus is large in the first $3$ years and it is very close to the upper bound in year 4: $b_4=8819.06$